Finite Group Databasebeta

← #12296

Subgroups — #12296

Order 256.

139 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 139 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3261C211yesmaximal, characteristic
2#2236C211yesmaximal, characteristic
3#2235C211yesmaximal, characteristic
4#2089C211yesmaximal, characteristic
5#1350C211yesmaximal, characteristic
6#1350C211yesmaximal, characteristic
7#1323C211yesmaximal, characteristic
8#541—21no—
9#479—21no—
10#479—21no—
11#470—21no—
12#467C2211yescharacteristic
13#467C2211yescharacteristic
14#462—21no—
15#462—21no—
16#428—21no—
17C2×#107C2211yescharacteristic
18#424C2211yescharacteristic
19#367—21no—
20#366—21no—
21#358—21no—
22#358—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#343—21no—
26#341C2211yescharacteristic
27C2×Q16—21no—
28C2×Q16D411yescharacteristic
29C2×Q16D411yescharacteristic
30C2×M16—21no—
31C2×M16D411yescharacteristic
32C22×C8C2311yescharacteristic
33C4⋊Q8—21no—
34#126—41no—
35#126—41no—
36#125—21no—
37#125—41no—
38#125—41no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43#117D411yescharacteristic
44C2×C4⋊C4—21no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107—41no—
50#107D411yescharacteristic
51#107D411yescharacteristic
52#106—21no—
53#106—41no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56#103—21no—
57#103—21no—
58#103—21no—
59#103—21no—
60C2×Q8—21no—
61C2×Q8—21no—
62C22×C4—41no—
63C22×C4C2×D411yescharacteristic
64Q16—41no—
65Q16—41no—
66Q16—81no—
67M16—22no—
68M16—41no—
69M16—41no—
70C2×C8—21no—
71C2×C8—41no—
72C2×C8—41no—
73C2×C8C2×D411yescharacteristic
74C2×C8C2×D411yescharacteristic
75C4⋊C4—21no—
76C4⋊C4—21no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4D811yescharacteristic
88C4⋊C4D811yescharacteristic
89C22⋊C4—21no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C42—21no—
93C42—41no—
94C42—41no—
95C16—41no—
96C16—41no—
97C23AGL(1,ℤ/8)11yescharacteristic
98Q8—41no—
99Q8—41no—
100Q8—41no—
101Q8—41no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—81no—
111C2×C4—161no—
112C2×C4#12011yescharacteristic
113C2×C4C2×D811yescharacteristic
114C8—21no—
115C8—21no—
116C8—42no—
117C8—41no—
118C22—21no—
119C22—21no—
120C22—21no—
121C22#44711yescharacteristic
122C4—21no—
123C4—41no—
124C4—41no—
125C4—81no—
126C4—81no—
127C4—81no—
128C4—81no—
129C4—161no—
130C4#45611yescharacteristic
131C4#45811yescharacteristic
132C2—41no—
133C2#161011yescharacteristic
134C2#219311yescharacteristic
135C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radicalC23AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#32612
#22362
#22352
#20892
#13502
#13232